{"id":"https://openalex.org/W7134122837","doi":"https://doi.org/10.1007/s11083-026-09731-6","title":"Poset Saturation of Unions of Chains","display_name":"Poset Saturation of Unions of Chains","publication_year":2026,"publication_date":"2026-03-07","ids":{"openalex":"https://openalex.org/W7134122837","doi":"https://doi.org/10.1007/s11083-026-09731-6"},"language":"en","primary_location":{"id":"doi:10.1007/s11083-026-09731-6","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s11083-026-09731-6","pdf_url":"https://link.springer.com/content/pdf/10.1007/s11083-026-09731-6.pdf","source":{"id":"https://openalex.org/S121371512","display_name":"Order","issn_l":"0167-8094","issn":["0167-8094","1572-9273"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Order","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s11083-026-09731-6.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5083025154","display_name":"Shengjin Ji","orcid":"https://orcid.org/0000-0002-5345-9953"},"institutions":[{"id":"https://openalex.org/I119203015","display_name":"Shandong University of Technology","ror":"https://ror.org/02mr3ar13","country_code":"CN","type":"education","lineage":["https://openalex.org/I119203015"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Shengjin Ji","raw_affiliation_strings":["School of Mathematics and Statistics, Shandong University of Technology, Zibo, 255000, Shandong, China"],"affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics, Shandong University of Technology, Zibo, 255000, Shandong, China","institution_ids":["https://openalex.org/I119203015"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5124358948","display_name":"Bal\u00e1zs Patk\u00f3s","orcid":null},"institutions":[{"id":"https://openalex.org/I127817813","display_name":"Alfr\u00e9d R\u00e9nyi Institute of Mathematics","ror":"https://ror.org/03vw74f64","country_code":"HU","type":"facility","lineage":["https://openalex.org/I127817813","https://openalex.org/I7597260"]}],"countries":["HU"],"is_corresponding":false,"raw_author_name":"Bal\u00e1zs Patk\u00f3s","raw_affiliation_strings":["HUN-REN Alfr\u00e9d R\u00e9nyi Institute of Mathematics, Re\u00e1ltanoda utca 13-15, Budapest, 1053, Hungary"],"affiliations":[{"raw_affiliation_string":"HUN-REN Alfr\u00e9d R\u00e9nyi Institute of Mathematics, Re\u00e1ltanoda utca 13-15, Budapest, 1053, Hungary","institution_ids":["https://openalex.org/I127817813"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5097353548","display_name":"Erfei Yue","orcid":"https://orcid.org/0009-0002-3979-7857"},"institutions":[{"id":"https://openalex.org/I106118109","display_name":"E\u00f6tv\u00f6s Lor\u00e1nd University","ror":"https://ror.org/01jsq2704","country_code":"HU","type":"education","lineage":["https://openalex.org/I106118109"]},{"id":"https://openalex.org/I127817813","display_name":"Alfr\u00e9d R\u00e9nyi Institute of Mathematics","ror":"https://ror.org/03vw74f64","country_code":"HU","type":"facility","lineage":["https://openalex.org/I127817813","https://openalex.org/I7597260"]}],"countries":["HU"],"is_corresponding":true,"raw_author_name":"Erfei Yue","raw_affiliation_strings":["HUN-REN Alfr\u00e9d R\u00e9nyi Institute of Mathematics, Re\u00e1ltanoda utca 13-15, Budapest, 1053, Hungary","Institute of Mathematics, E\u00f6tv\u00f6s Lor\u00e1nd University, P\u00e1zm\u00e1ny P\u00e9ter s\u00e9t\u00e1ny 1/C, Budapest, 1117, Hungary"],"affiliations":[{"raw_affiliation_string":"HUN-REN Alfr\u00e9d R\u00e9nyi Institute of Mathematics, Re\u00e1ltanoda utca 13-15, Budapest, 1053, Hungary","institution_ids":["https://openalex.org/I127817813"]},{"raw_affiliation_string":"Institute of Mathematics, E\u00f6tv\u00f6s Lor\u00e1nd University, P\u00e1zm\u00e1ny P\u00e9ter s\u00e9t\u00e1ny 1/C, Budapest, 1117, Hungary","institution_ids":["https://openalex.org/I106118109"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5097353548"],"corresponding_institution_ids":["https://openalex.org/I106118109","https://openalex.org/I127817813"],"apc_list":{"value":2290,"currency":"EUR","value_usd":2890},"apc_paid":{"value":2290,"currency":"EUR","value_usd":2890},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.91700242,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":null,"biblio":{"volume":"43","issue":"1","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11329","display_name":"Limits and Structures in Graph Theory","score":0.42719998955726624,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11329","display_name":"Limits and Structures in Graph Theory","score":0.42719998955726624,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.267300009727478,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11476","display_name":"Graph theory and applications","score":0.03530000150203705,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/partially-ordered-set","display_name":"Partially ordered set","score":0.8973000049591064},{"id":"https://openalex.org/keywords/bijection","display_name":"Bijection","score":0.8593000173568726},{"id":"https://openalex.org/keywords/saturation","display_name":"Saturation (graph theory)","score":0.5372999906539917},{"id":"https://openalex.org/keywords/existential-quantification","display_name":"Existential quantification","score":0.41119998693466187},{"id":"https://openalex.org/keywords/ordered-set","display_name":"Ordered set","score":0.4025000035762787}],"concepts":[{"id":"https://openalex.org/C180645754","wikidata":"https://www.wikidata.org/wiki/Q474715","display_name":"Partially ordered set","level":2,"score":0.8973000049591064},{"id":"https://openalex.org/C24424167","wikidata":"https://www.wikidata.org/wiki/Q180907","display_name":"Bijection","level":2,"score":0.8593000173568726},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8065999746322632},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6554999947547913},{"id":"https://openalex.org/C9930424","wikidata":"https://www.wikidata.org/wiki/Q7426587","display_name":"Saturation (graph theory)","level":2,"score":0.5372999906539917},{"id":"https://openalex.org/C136979486","wikidata":"https://www.wikidata.org/wiki/Q773483","display_name":"Existential quantification","level":2,"score":0.41119998693466187},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4081999957561493},{"id":"https://openalex.org/C2985459651","wikidata":"https://www.wikidata.org/wiki/Q474715","display_name":"Ordered set","level":2,"score":0.4025000035762787},{"id":"https://openalex.org/C199185054","wikidata":"https://www.wikidata.org/wiki/Q552299","display_name":"Chain (unit)","level":2,"score":0.3001999855041504},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.25850000977516174}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1007/s11083-026-09731-6","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s11083-026-09731-6","pdf_url":"https://link.springer.com/content/pdf/10.1007/s11083-026-09731-6.pdf","source":{"id":"https://openalex.org/S121371512","display_name":"Order","issn_l":"0167-8094","issn":["0167-8094","1572-9273"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Order","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1007/s11083-026-09731-6","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s11083-026-09731-6","pdf_url":"https://link.springer.com/content/pdf/10.1007/s11083-026-09731-6.pdf","source":{"id":"https://openalex.org/S121371512","display_name":"Order","issn_l":"0167-8094","issn":["0167-8094","1572-9273"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Order","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7134122837.pdf","grobid_xml":"https://content.openalex.org/works/W7134122837.grobid-xml"},"referenced_works_count":16,"referenced_works":["https://openalex.org/W190512601","https://openalex.org/W1981663372","https://openalex.org/W2034962669","https://openalex.org/W2068795011","https://openalex.org/W2502085572","https://openalex.org/W2576813251","https://openalex.org/W2909784596","https://openalex.org/W3009260386","https://openalex.org/W3009504246","https://openalex.org/W4221031172","https://openalex.org/W4390104109","https://openalex.org/W4396906848","https://openalex.org/W4400197655","https://openalex.org/W4408890044","https://openalex.org/W4410888628","https://openalex.org/W6902852323"],"related_works":[],"abstract_inverted_index":{"Abstract":[0],"A":[1],"family":[2,166],"$$\\varvec{{\\mathcal":[3,167],"{G}}}$$":[4,56],"<mml:math":[5,22,57,82,104,134,171,200,216],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">":[6,23,58,83,105,135,172,201,217],"<mml:mrow>":[7,24,25,28,31,34,37,40,43,59,60,63,66,69,72,84,85,88,91,106,107,110,115,118,121,136,137,143,146,149,152,155,173,174,177,181,184,202,218,219,222,226,229,235,238],"<mml:mi>G</mml:mi>":[8,73,220],"</mml:mrow>":[9,27,30,33,36,39,42,45,46,62,65,68,71,74,75,87,90,93,94,109,114,117,120,125,126,142,145,148,151,154,157,158,176,179,183,188,190,204,221,224,228,233,237],"</mml:math>":[10,47,76,95,127,159,191,205],"of":[11,17,164,198],"sets":[12],"is":[13,160],"a(n":[14],"induced)":[15],"copy":[16,197],"a":[18,51,165],"poset":[19],"$$\\varvec{P}\\varvec{=}\\varvec{(}\\varvec{P}\\varvec{,}\\varvec{\\leqslant":[20],"}\\varvec{)}$$":[21],"<mml:mi>P</mml:mi>":[26,35,67,153,203],"<mml:mo>=</mml:mo>":[29],"<mml:mo>(</mml:mo>":[32,111,122,144],"<mml:mo>,</mml:mo>":[38,150],"<mml:mo>\u2a7d</mml:mo>":[41,89],"<mml:mo>)</mml:mo>":[44,113,124,156],"if":[48,97,100],"there":[49],"exists":[50],"bijection":[52],"$$\\varvec{b}\\varvec{:}\\varvec{P}\\varvec{\\rightarrow":[53],"}":[54,80,102,169,211,213],"\\varvec{{\\mathcal":[55,214],"<mml:mi>b</mml:mi>":[61,108,119],"<mml:mo>:</mml:mo>":[64],"<mml:mo>\u2192</mml:mo>":[70],"such":[77],"that":[78,192],"$$\\varvec{p}\\varvec{\\leqslant":[79],"\\varvec{q}$$":[81],"<mml:mi>p</mml:mi>":[86,112],"<mml:mi>q</mml:mi>":[92,123],"holds":[96],"and":[98],"only":[99],"$$\\varvec{b}\\varvec{(p)}\\varvec{\\subset":[101],"\\varvec{b}\\varvec{(q)}$$":[103],"<mml:mo>\u2282</mml:mo>":[116],".":[128],"The":[129],"induced":[130],"saturation":[131],"number":[132],"$$\\varvec{\\textrm{sat}^*}\\varvec{(}\\varvec{n}\\varvec{,}\\varvec{P}\\varvec{)}$$":[133],"<mml:msup>":[138,180,225],"<mml:mtext>sat</mml:mtext>":[139],"<mml:mo>\u2217</mml:mo>":[140],"</mml:msup>":[141,189,234],"<mml:mi>n</mml:mi>":[147,186,231],"the":[161],"minimum":[162],"size":[163],"{F}}}\\varvec{\\subseteq":[168],"\\varvec{2}^{\\varvec{[n]}}$$":[170],"<mml:mi>F</mml:mi>":[175],"<mml:mo>\u2286</mml:mo>":[178],"<mml:mn>2</mml:mn>":[182,227],"<mml:mo>[</mml:mo>":[185,230],"<mml:mo>]</mml:mo>":[187,232],"does":[193],"not":[194],"contain":[195],"any":[196,209],"$$\\varvec{P}$$":[199],",":[206],"but":[207],"for":[208],"$$\\varvec{G}\\varvec{\\in":[210],"\\varvec{2}^{\\varvec{[n]}}\\varvec{\\setminus":[212],"{F}}}$$":[215],"<mml:mo>\u2208</mml:mo>":[223],"<mml:mo>\\</mml:mo>":[236],"<":[239]},"counts_by_year":[],"updated_date":"2026-03-10T14:07:55.174380","created_date":"2026-03-08T00:00:00"}